SM1005 User Guide_0314.pdf

SM1005 Loading and Buckling of Struts

User Guide

© TecQuipment Ltd 2014 Do not reproduce or transmit this document in any form or by any means, electronic or mechanical, including photocopy, recording or any information storage and retrieval system without the express permission of TecQuipment Limited. TecQuipment has taken care to make the contents of this manual accurate and up to date. However, if you find any errors, please let us know so we can rectify the problem. TecQuipment supplies a Packing Contents List (PCL) with the equipment. Carefully check the contents of the package(s) against the list. If any items are missing or damaged, contact TecQuipment or the local agent.

DB/AD/bs/0314


Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

The Main Parts and the Load Meter (Display) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Struts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Deflection Indicator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eccentric End Fittings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weights and Hangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adjustable Adjustable and Removable Removable Fixings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Versatile Data Acquisition System (VDAS®) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 4 4

Technical Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 6 6 7

Young’s Modulus Mod ulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Noise Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Installation and Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Location and Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Electrical Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Connections (including VDAS®) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Notation, Useful Equations and Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Useful Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Theory of Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Safety. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Useful Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experiment Experiment 1 - Deflection of a Simply Supported Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experiment 2 - Stiffness (Young's Modulus) of the Strut Materials . . . . . . . . . . . . . . . . . . . . Further Experiments with Beams - Beam Deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experiment 3 - The Deflected Shape of a Strut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experiment 4 - The Euler Buckling Load using Pinned-end Struts . . . . . . . . . . . . . . . . . . . . Experiment 5 - Comparing Buckling loads with End Conditions . . . . . . . . . . . . . . . . . . . . . . Experiment Experiment 6 - The Southwell Plot and the Buckling Load . . . . . . . . . . . . . . . . . . . . . . . . . . Experiment 7 - The Southwell Plot and Eccentricity of Loading . . . . . . . . . . . . . . . . . . . . . . Further Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 27 31

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Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

Experiment 1 - Deflection of a Simply Supported Beam . . . . . . . . . . . . . . . . . . . . . . . . . Experiment 2 - Stiffness (Young's Modulus) of the Strut Materials . . . . . . . . . . . . . . . Experiment 3 - The Deflected Shape of a Strut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experiment 4 - The Euler Buckling Load using Pinned-end Struts. . . . . . . . . . . . . . . . Experiment 5 - Comparing Buckling loads with End Conditions . . . . . . . . . . . . . . . . . Experiment 6 - The Southwell Plot and the Buckling Load . . . . . . . . . . . . . . . . . . . . . . . Experiment 7 - The Southwell Plot and Eccentricity of Loading . . . . . . . . . . . . . . . . . .

49 50 52 54 55 55 56

Useful Textbooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

Maintenance, Spare Parts and Customer Care . . . . . . . . . . . . . . . . . . . . . . .

61

Maintenance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Spare Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Customer Care . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62


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User Guide


User Guide Introduction

Figure 1 Loading and Buckling of Struts (SM1005)

This product works with VDAS ®

Engineers learning about structures need to know how to predict the effects of compression forces on struts. They can use this information to decide the right type and thickness of materials for their own designs. TecQuipment’s Loading and Buckling of Struts shows students how struts of different sizes, materials and cross-section deflect and buckle under load. This mimics struts in real applications, such as roof supports in buildings or parts of frameworks in a structure. The equipment includes ten struts of different metals, lengths and cross-sections. You can also buy the additional pack of struts (SM1005a) for more experiment with struts of different materials, including wood and glass fibre. The apparatus teaches students about the most important factors that affect how well a strut can resist a buckling load. These include: • Different end conditions (how you hold or clamp the ends of a strut). • Eccentricities of loading. • The material and dimensions of the strut. It also includes parts to allow basic beam bending tests, to help introduce students to bending theory. TecQuipment Ltd

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Description Deflection indicator

Load measuring end

Loading end with Hand wheel

Knife-edge support

Weights and parts to apply loads for beam bending and side loads Knife-edge support

Load Meter

Standard set of different material and cross-section struts (supplied).

Standard set of different length struts (supplied).

Figure 2 Parts of the Loading and Buckling of Struts (SM1005)

The Main Parts and the Load Meter (Display) The main part of the Loading and Buckling of Struts (SM1005) is a precision frame with adjustable fee t. The frame has slots to hold the loading end and the load measuring end. The slots allow you to adjust the distance between the two ends to fit different size struts. The frame also has slots on its front and back, each above a scale. These slots allow you to fit two adjustable knife-edge supports and the deflection indicator, for simple beam bending tests. The loading end has a hand wheel that turns a thread to give a compression force on the end of a strut. The load measuring end has a load sensor connected with a unique mechanism. This mechani sm allows the sensor to measure the axial force (buckling load) on the strut, but ignores any bending (rotating) forces. The load sensor connects to a separate Load Meter (Display) that shows the axial force on the strut. The Load Meter has a socket for connection to TecQuipment’s optional VDAS®. VDAS® allows data acquisition from this equipment, with the use of a suitable computer (not supplied). The Load Display has two buttons. Press one button to zero the load display before you take any readings. The other button sets the display to hold a peak value of the force. This is useful to help find the maximum buckling load of a strut. To set the peak hold, press and hold this bu tton for a few seconds. A small symbol appears in the display to show that the peak hold is working. The display now shows two readings, one is actual load, the other is the peak (maximum) value that the display has measured du ring your experiment. Press and hold the peak hold button to cancel the peak hold.

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The Struts

Figure 3 Each Standard Strut Has a Number Stamped Near Its End and Some have Extra Holes

Included as standard are ten struts. Each standard strut has a number stamped near one end (see Figure 3). Some struts also have extra holes to accept the spec ial Eccentric End Fittings. Six struts are the same material, thickness and width, but different lengths, to compare the effect of strut length. The longer struts also allow for length ‘lost’ in the end fixings, so you can compare them with shorter struts. Four struts are of different materials, thickness and length. The standard struts are all solid cross-section metal struts. For extra experiments you can also buy the SM1005A pack of optional struts. This includes struts of different material and cross-section, including wooden struts, a compound strut and struts with angle and channel cross-section. See Technical Details on page 9 for more details.

The Deflection Indicator

Figure 4 The Deflection Indicator Fits on Two Different Holders for Different Experiments

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A digital deflection indicator measures how much the struts deflect (bend). It mounts on an L-shape holder for strut experiment on the top of the base. It also mounts on a flat holder on the side of the base for beam bending experiments. Supplied with the equipment is a cable to connect the indicator to TecQuipment’s optional VDAS®. An adjustable weight hanger, knife edge hanger and pulley allow students to apply a light biasing load or side load to the strut under test. This is good for side load tests, and to prove any theo ry about struts that are already curved. Students also use the weight hanger and knife edge hanger to apply a load for beam bending tests.

Eccentric End Fittings

Figure 5 The Eccentric End Fittings

Also included are two special end fittings to help apply out-of-centre loads to the struts, for tests on eccentric loading. They have two sides, to allow two different out-of-ce ntre (eccentric) loading distances on a standard strut (see Figure 6).

Figure 6 Out of Centre Loading with the Eccentric End Fittings

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Weights and Hangers

Weight Hanger and Weights

Knife Edge Weight Hanger

Cord and Knife Edge Weight Hanger

Pulley assembly

Figure 7 Weights and Weight Hangers

Included with the equipment are some extra parts. These are: • A pulley assembly that fits in one of the deflection indicator holders. You use this with the cord and knife edge weight hanger to apply load for side load tests. • A Weight Hanger, Weights and a second Knife Edge Weight Hanger. You use these to apply a load for beam tests and for side load tests.

Adjustable and Removable Fixings

Figure 8 To Remove Fixings

TecQuipment put the fixings in the slots of the frame to suit a standard arrangement. However, if you need to move the deflection holder or the beam supports to the opposite side of the frame, you can use a steel rule to remove the fixings (see Figure 8). Carefully insert them back into the slots where yo u need them.

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Versatile Data Acquisition System (VDAS ® )

Figure 9 The VDAS® Hardware and Software

TecQuipment’s VDAS® is an optional extra for the Loading and Buckling of Struts. It is a two-part product (Hardware and Software) that will: • automatically log data from your experiments • automatically calculate data for you • save you time • reduce errors • create charts and tables of your data • export your data for processing in other software

NOTE

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You will need a suitable computer (not supplied) to use TecQuipment’s VDAS®.

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Technical Details

Item

Details

Nett Dimensions

Main Unit: 1350 mm long x 500 mm front to back x 500 mm high Digital Load Display: 170 mm x 60 mm x 200 mm

Nett Weight

Main Unit: 24 kg Digital Load Display: 1.4 kg

Electrical Supply (for the Digital Load Display Power Supply

Input 90 VAC to 264 VAC 50 Hz to 60 Hz at 1A Output 12 VDC at 5 A Centre Positive

Fuse

No fuses fitted.

Operating Environment

Indoor (laboratory) Altitude up to 2000 m Overvoltage category 2 (as specified in EN61010-1). Pollution degree 2 (as specified in EN61010-1).

Maximum Load Capacity of Load Measurement Unit

2000 N

Maximum strut length allowable

Approximately 800 mm in fixed - fixed ends.

Eccentric End Fittings

Offset the loading centre of a 3 mm strut by 5 mm and 7.5 mm.

Standard Struts Nominal Dimensions

Strut 1 - Steel, 20 mm x 3 mm x 750 mm Strut 2 - Steel, 20 mm x 3 mm x 700 mm Strut 3 - Steel, 20 mm x 3 mm x 650 mm Strut 4 - Steel, 20 mm x 3 mm x 625 mm Strut 5 - Steel, 20 mm x 3 mm x 600 mm Strut 6 - Steel, 20 mm x 3 mm x 550 mm Strut 7 - Brass, 19 mm x 4.8 mm x 750 mm Strut 8 - Aluminium, 19 mm x 4.8 mm x 750 mm Strut 9 - Steel, 15 mm x 4 mm x 750 mm Strut 10 - Steel, 10 mm x 5 mm x 750 mm

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Item Optional Struts Nominal Dimensions

Details Strut A - Hardwood (Mahogany), 20 mm x 6 mm x 550 mm with steel knife edge inserts. Strut B - Plywood (Marine Ply), 20 mm x 6 mm x 550 mm with steel knife edge inserts. Strut C - Glass Fibre, 20 mm x 5.5 mm x 550 mm with steel knife edge inserts. Strut D - Moulded Brass (Extruded brass), D shape, 19 mm x 4.5 mm x 550 mm Strut E - Steel Compound, Two steel struts, bolted together. Both 13 mm x 3 mm. One longer (650 mm) with knife edge ends, the other 640 mm. Strut F - Aluminium channel, 13 mm x 13 mm and 1.5 mm thick wall x 750 mm with steel end fittings. Strut G - Aluminium angle, 13 mm x 13 mm and 1.5 mm thick wall x 750 mm with steel end fittings. Strut H - Aluminium angle, 13 mm x 13 mm and 1.5 mm thick wall x 750 mm with steel end fittings. Strut I - Steel Rectangular, 13 mm x 6.4 mm x 650 mm. Strut J - Steel Round, 6.4 mm diameter x 650 mm.

Young’s Modulus

Young’s Modulus (Nominal)

Material Steel

207 GN.m-2 (207 GPa)

Aluminium

69 GN.m-2 (69 GPa)

Brass

105 GN.m-2 (105 GPa)

Noise Levels The noise levels recorded at this apparatus are less than 70 dB (A).

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Installation and Assembly The terms left, right, front and rear of the apparatus refer to the operators’ position, facing the unit.

NOTE

• A wax coating may have been applied to parts of this apparatus to prevent corrosion during transport. Remove the wax coating by using paraffin or white spirit, applied with either a soft brush or a cloth. • Follow any regulations that affect the installation, operation and maintenance of this apparatus in the country where it is to be used.

Location and Assembly Use the Loading and Buckling of Struts in a clean, well-lit laboratory or classroom type area. Put it on the top of a solid, level workbench.

WARNING

When assembled, the equipment weighs more than 20 kg. Ask an assistant to help you move it by holding its support legs.

The Loading and Buckling of Struts uses a bench area of 1350 mm x 500 mm. If you are to use the optional VDAS®, allow room nearby for a computer. Obey the manufacturer’s instructions (supplied) to fit the batteries to the digital deflection indicator.

Assembly 1. Fit the four legs to the main part of the frame, then fit the scale bar to the legs, underneath the base unit (see Figure 10).

Figure 10 Fit the Legs to the Frame, then fit the Scale Bar

2. Adjust the feet on the legs until the frame is level. 3. Connect the power supply for the Load Display to the electrical supply, then connect the sensor cable from the load measuring end to the socket at the back of the Load Display. 4. Refer to the experiment for more assembly details.

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Electrical Connection Use the cable supplied to connect the Load Display power supply to a single-phase electrical supply. Connect the apparatus to the supply through a plug and socket. The apparatus must be connected to earth.

WARNING

The mains supply connector at the Power Supply is its mai ns disconnect device. Make sure it is always easily accessible.

These are the colours of each individual conductor:

GREEN AND YELLOW:

EARTH E OR

BROWN:

LIVE

BLUE:

NEUTRAL

Connections (including VDAS ® ) If you are to use the optional VDAS® with the Loading and Buckling of Struts, read the VDAS® User Guide and connect the Loading and Buckling of Struts to the VDAS-B Inter face and computer as shown in Figure 11. Included with the SM1005 is the SPC cable that connects the digital deflec tion indicator to the optional VDAS-B Interface.

Digital Indicator

Force Sensor Load Cell

To mains electrical supply

To mains electrical supply

Mains to low voltage power supply

Mains to low voltage power supply DIGITAL INPUTS

DIGITAL OUTPUT 1

DIGITAL OUTPUT INPUTSOCKET1

2 RS232

POWER

COMMS 3

INPUTSOCKET2

OR 4

HUBBOARD

DTIINPUTBOARD

ANALOGUE IN PUT BOARD

VERSATILE DATAACQUISITION SYSTEM

Digital Load Meter

Optional VDAS Hardware

Figure 11 Connection to VDAS®

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Notation, Useful Equations and Theory This section only gives the basic information needed to do the experiments. For full theory, refer to the textbooks listed in Maintenance, Spare Parts and Customer Care on page 61.

Notation

Symbol

Definition

Units

Force or a ‘Prop Force’

N

‘Effective Length’

m (or mm where stated)

L

Total length


A

Area of cross-section

m2 (or mm2 where stated)

σ

Normal Stress

N.m-2

ε

Direct Strain

Strain or micro-strain

K

Column effective length factor

-

I

‘Area moment of inertia’ or ‘Second moment of area’

m4 (or mm4 where stated)

E

Young’s Modulus

Pa (or N.m -2 or kN.mm-2 where stated)

x

Distance along a strut


y

Displacement or deflection at a position along a strut


yo

Initial displacement or deflection at a position along a strut


yc

Displacement or deflection at the centre of (half-way along) a strut


yco

Initial displacement or deflection at the centre of (halfway along) a strut


F or P l

Conversions Second Moment of Area: To convert mm4 into m4, multiply by 10 -12 (1 mm 4 = 1 x 10-12 m4) Young’s Modulus: 1 GPa = 1 GN.m -2 = 1 kN.mm-2

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Useful Equations Forc e (F), Prop Fo rce (P) and L oad You use a set of weights to apply a load for beam tests, but in your calculations, you must use the force caused by the weights that make the load. From Newton’s Law of F = ma: Force (N) = Load (kg) x g (9.81 m.s-2) Many standard equations use the letter F for force. This works for the beam experiments, but for strut experiments, the force is a prop force , applied to the end of the strut. So, this theory uses the letter ( P ) for this force.

Yo u n g ’ s M o d u l u s ( Ε ) This is a ratio of the stress divided by the strain on a material. An English physicist - Thomas Young discovered it. It is a measure of the stiffness of a material (a stiffer material has a higher value of Young’s Modulus). It is found by the equation:

E =

σ --ε

S ec o n d M o m e n t o f A r e a The second moment of area for a rectangular cross-section beam is: 3

I =

bd

-------12

(1)

This is the minimum second moment of area for the strut or beam with a load applied so that it bends in its weakest direction.

NOTE

d b Figure 12 Cross-section of a Beam

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EI Flexural Rigidity ( ) The second moment of area ( I ) of a beam or strut links to its dimensional strength. The Young’s Modulus ( E ) of a beam or strut links to the strength of its material (while being used in its elastic region). The product of these two values ( E x I ) gives a measure of the beam’s flexural rigidity. A higher flexural rigidity means that you need more force to bend or flex a beam of a known length than an identical beam, that has a lower flexural rigidity. The length of the beam and other factors decide the actual force need to bend the beam.

Beam Bending Theory Deflection of a Beam on Simple Supports

Figure 13 Beam on Two Supports

Figure 13 shows a simply supported beam on two supports, loaded at exactly the mid-point. For this arrangement, the theoretical deflection at the mid-point ( yc) is: 3

c =

FL -----------48 EI

(2)

Stiffness of a beam and Young’s Modulus As shown later in this guide, struts are not usually perfect. They may not be perfectly straight. They may not have a constant cross-sectional area along their length, or may not have a textbook value Young’s Modulus. This can give errors in your results. To reduce these errors, you can do accurate stiffness tests on each strut to find its true Young’s Modulus. To make things simple, you can use the simply supported beam test and its theory. To do this, you can rearrange Equation 2 to give:

F

=

48 y I E × -----------c-3 L

The right hand side of this equation relates to the beam’s stiffness. You need more force to bend a stiffer beam. This equation is also in the form y = mx. From this, a chart of F against 48 yc I / L3 will give a straight line, with the gradient E . This will be a good average measurement of the Young’s Modulus along the beam’s test length.

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Figure 14 How to Use Your Results to Find Young’s Modulus

Beam Bending Moment

Figure 15 Beam Bending Theory - Bending Moment and Deflection

Figure 15 shows a cantilever beam. This is similar to a strut with one fixed end and one free end. Bending Moment ( M ) = - F x ( L - x) Also, from the differential equation of bending: 2

M

User Guide

=

16

d y EI -------2 dx

(3)

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Theory of Buckling F ai l u r e o f L o n g a n d S h o r t C o l u m n s

Axial Compressive Load

A Long Column Axial Compressive Load

A Short Column



Figure 16 Long and Short Columns

Columns are parts of structures that resist compressive axial loads (usually vertical loads). Stanchions are upright (usually metal) columns in buildings. Struts are the smaller parts (members) that resist compression in trusses and frames. Columns can be either long ( slender ) or short (fat ) (see Figure 16). When compressed by too much axial load, long, slender columns fail by suddenly bending out of line (for example - a plastic ruler). They become ‘unstable’ and ‘buckle’ at a maximum or ‘critical (buckling ) load’ . Short, fat columns fail in several ways, mostly determined by the material they are made from (for example - concrete crushes and mild steel yields). In reality, most columns are ‘intermediate’ - they fail by a combination of effects, where bending starts a material failure. However, this equipment examines a single effect - how slender struts or columns fail by buckling.

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The Fou r Main Factors that Affect Bu cklin g The words ‘long’ and ‘slender’ suggest the length ( L) and cross-section of a strut affect its buckling load. However, the cross-sectional shape also affects the buckling load. The ‘second moment of area ’ ( I ) is a measure of both the cross-sectional area and shape. It affects the stiffness of the strut. In addition, the elastic bending of a strut depends on the Young’s Modulus ( E ) of the material it is made from. So, the E value affects the buckling load. Finally, and perhaps slightly less obviously, the buckling load of a slender column depends on its end conditions . Firmly clamped or ‘fixed ends’ help the column to withstand higher buckling loads than a column that has end fixings that are less rigid.

Eulers Maximu m (Critical) Ax ial Bu cklin g Lo ad and ‘Effective Length’ A Swiss mathematician - Leonhard Euler, created a formula that predicts the maximum (critic al) axial buckling load ( P cr ) of a strut. 2

P cr

π EI

(4)

= --------------

( KL )

2

Where K is an ‘effective length factor’ - determined by how you fix the ends of the strut. It is the ratio of the ‘effective length’ (l ) between two points, to the overall length ( L) of the strut. The equation again shows that the Young’s Modulus and cross-sectional dimensions (second moment of area) affect the maximum buckling load. It also shows that the buckling load varies linearly with these quantities. This allows you to see that, for example, a steel strut with an E value of 200 GPa should buckle at twenty times the load of an equivalent wooden strut, if the wood has an E value of only 10 GPa.

Results for struts of different lengths

X X

Shorter struts

2

1/ L

X X

Longer struts

Buckling Load Figure 17 Length Squared Against Buckling Load

The equation also shows that buckling load is inversely proportion al to the square of a column’s length. A chart of 1/ L2 against buckling load will be linear (see Figure 17). This proves that longer columns have lower buckling loads, but also shows that buckling load is sensitive to column length (doubling the length will quarter the buckling load).

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Figure 18 shows that the way you fix a strut decides its effective length. A strut with one fixed end has an effective length of 0.7 of its total length. A strut with two fixed ends has an e ffective length of 0.5 of its total length. This assumes that you fix the ends firmly - any movement in the ends will affect your calculations.

Figure 18 Eulers Equations for Different Strut End Conditions

Shape and Deflection of Bu ckled Struts

Buckled shape = half a complete sine wave = effective length y P cr

x

P cr

Figure 19 Shape of a Pin-ended Strut Under Load

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Figure 19 shows the buckled shape of a Pin-ended strut that is initially straight . The shape is symmetrical (half a sine wave), and its bending moment:

M

= – P cr ×

y (at a distance x and a deflection y)

Also, from the standard differential equation of bending: 2

M

=

d y EI -------2 dx

l = 0.5 L = half a complete c ycle of a sine wave = effective length

P cr

P cr

L = one complete c ycle of a sine wave

Figure 20 Shape of a Fixed end Strut Under Load

Figure 19 shows that a pinned-end strut under load buckles so that it forms a symmetrical curve (half cycle sine wave). Its effective length is the full length of the strut. Figure 20 shows that a fixed-ended strut buckles so that it forms a full sine wave, but its effective length (that corresponds with the pinnedend strut) is only half its entire length. So, we can consider a fixed-end strut to have half the effective length of the pinned-end strut . Figure 18 gives the Euler equation for the fixed end strut. It shows that a fixed-end strut has four times the buckling load of an equivalent pin-ended strut.

Figure 21 Fixed and Pin-ended Strut

For a strut that is fixed at one end but pinned at the other (see Figure 21), it is not possible to predict its effective length precisely by looking at it, but bending theory shows mathematically that it is approximately 0.7 L , so that its buckling load is slightly greater than twice (2.04 times, more accurately ) the buckling load of the pinned-end strut. It’s shape is approximately 1/2 of a sine wave.

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From this theory, if the pinned ends condition has a buckling load of 1 kN, the fixed and pinned end condition has a buckling load twice this (2 kN). The fixed ends condition has a buckling load of four times this (4 kN).

Figure 22 Buckled Shape of an Initially Curved Strut

Figure 22 shows the buckled shape of a strut that already has a displacement at its central posi tion. This curve is symmetrical (a half cycle of a sine wave). The deflection equation fo r any point along the initial curve is:

y o

π x y co sin  ------  L 

=

(5)

The bending equation (3) becomes: 2

d y EI -------- = 2 dx

2

– P ( y +

y o ) OR

d y

P EI

P EI

π x L

-------- + ----- y = – ------ y co sin ------

dx

2

The solution to this equation is: 2

=

Where µ

2

µ y co

π x A cos µ x + B sin µ x + ----------------- sin -----2 L 2 π ----- – µ 2 L

P EI

= -----

At the conditions y = 0, x = 0, L gives A = 0 and B = 0, so: 2

µ y co

π x L

= ----------------- sin ------ OR

π

2

----- – µ

L

2

2

yc

y co

= --------------------2

π EI

----------- – 1

PL

2

So

y c

y co

= ----------------

P cr

(6)

------- – 1

P

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Slenderness Ratio and Bu ckling Stress The radius of gyration, r , of a section is the distance from its centroid at which its area may be effectively considered to be concentrated. As stated earlier, the second moment of area ( I ) links with the cross-sectional shape and area of a beam or strut. It also links with the area and the radius of gyration, so that:

I

=

2

Ar

Substituting this in the Euler buckling equation gives: 2

P cr

2

π EI

π EA r

= -------------- = ----------------

( KL )

2

( KL )

2

(7)

As shown earlier, stress (σ) is the force divided by the area ( F/A). From this, the stress ( σcr ) at the buckling load is:

σ cr

P cr

(8)

= -------

A

Substituting with Equation 7, we get: 2

2

π Er

σ cr

= --------------

( KL )

2

(9)

and: 2

σ cr

π E

= ---------------

 KL  ------ r 

2

(10)

KL/r (or l/r in this theory) is the slenderness ratio of the strut. It is a measure of buckling resistance. Equation (10) shows that buckling stress is inversely related to the square of the ratio. From this, you should always use the minimum dimensions of your strut (for example - if you have a non-symmetric strut), to calculate the slenderness ratio. This is because a strut always buckles in the direction that matches the weakest dimension of the strut. However, of course the end fixings affect this as well.

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Effects of Imperfection s

Initially straight strut Critical (buckling) Load

Initially curved strut

Load

Deflection Figure 23 Comparison of Straight and Curved Strut

Equation (4) depends on three assumptions: • The strut has constant values of E and I , so that it is homogeneous (has constant material properties). • The strut is prismatic (has constant cross-section and therefore I value). • The strut is perfectly straight. In reality, none of these assumptions can be perfectly true, especially straightness. This is important. Remember that Figure 22 (an already curved strut) and its theory shows a strut with a known lack of initial straightness. It also shows that deflection increases rapidly as the compressive load approaches its critical value .

T h e S o u t h w e l l P l o t a n d ‘ E c c en t r i c i t y o f L o a d i n g ’ Figure 23 shows a comparison of the deflection under load of initially straight and curved struts. With the initially straight strut under perfect conditions, it only deflects when the load reaches the critical value. This gives a clear visual display of the point of buckling. For the initially curved strut, the gradual increase in deflection makes it difficult to see the point of buckling. In reality, struts are not perfect and most will gradually deflect as you apply load. To help with this, you can use a Southwell Plot . A rearranged version of Equation 6 gives this plot, so:

y c

TecQuipment Ltd

=

y P cr  ----c – y co  P 

23

(11)

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yc Gradient = P cr

yc P Intercept = y co

Figure 24 Southwell Plot

Equation 6 shows that a plot of yc against yc/ P will be linear (see Figure 24) and that its slope gives an approximate value of the critical load. Also, the intercept on the y-axis gives an approximate value for the original central out-of-straightness or ‘eccentricity of loading’ ( yco). This helps you to see how imperfect the strut is.

The Southwell Plot and Testing Struts During the experiments, you will find two important factors that make the Southwell Plot a useful tool for predicting the properties of a strut. 1. You do not need to test a real strut to its critical (buckling) load to create the Southwell Plot. This allows you to do tests without risk of damaging the struts. 2. For accurate results in some experiment, you need to do two tests and work out the average buckling load (due to the struts natural buckling direction). This is not necessary for the Southwell Plot.

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Experiments Safety

WARNING

WARNING

Never try to release a strut from its end fixings when it is under load. Always reduce the load force to zero before you change or adjust a strut.

If you do not use the equipment as described in these instructions, its protective parts may not work correctly.

Useful Notes • Do these experiments in order, the results and procedures of the earlier experiments help you to understand the later experiments. • The Euler theory assumes that the struts are perfectly straight, this will never be what happens in real life, and some of these experiments compare the theor y and reality. It is impossible to make a perfect strut, but TecQuipment take care to make sure the struts supplied are reasonably straight. • The struts will give good results unless someone accidently bends them past their elastic limit. The experiments only use the struts within their elastic limit, so you can reuse them. However, if you do not use the equipment correctly you may bend the struts too far, so take care as you reach the buckling load. • A strut is still useable, even if it has a slight curve. It is not useable if it has a sharp bend or ‘kink’. You may straighten a slight bend in a strut. To do this, carefully bend it back in the opposite direction by hand, or use a set of rollers (if you have them). This may affect some properties of the strut, but will not affect its bending and buckling properties.

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Experiment 1 - Deflection of a Simply Supported Beam Aims To verify the simple beam bending equation for a beam on two supports and show the background for the Euler formula.

Procedure 1. Create a blank results table, similar to Table 1. If you have VDAS®, select ‘Beam Experiments’. The software will create a table for you automatically when you start taking readings.

Second Moment of Area ( I ): Young’s Modulus ( E ):

Beam Material: Distance between supports ( L):

Load (g)

Force (N)

0

Measured Deflection (mm)

Measured Deflection (m)

FL3

Theoretical Deflection ( y)

0

100 200 300 400 500

Table 1 Blank Results Table

2. Loosen the fixings of the measuring end if necessary and move it to the end of the base (see Figure 25). 3. Fit the two knife edge supports to the front side of the main base, above the measurement scale, so they are exactly 600 mm apart (for example - set them to 200 mm and 800 mm). Make sure the sharp edge is upwards. If necessary, use a screwdriver or the thumbscrews to slide the fixings along in the slots so you can fit the thumbscrews (see Figure 26).

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Figure 25 Loosen the Fixings and Move the Measuring End Along to the End of the Base

Sharp edge upwards

Thumbscrew

Use a screwdriver or a thumbscrew to slide the fixings along.

Figure 26 Fit the Knife Edge Supports

4. Fit the flat plate digital deflection indicator holder to the front of the base, so that it is exactly midway between the two knife edge supports (for example - at 500 mm). 5. Find the 750 mm long steel specimen strut (number 1). Use an accurate vernier or micrometer and carefully measure the dimensions of the strut. Use these to calculate the second moment of area for the strut. If you have VDAS®, enter this value into the software. Also enter the material type and its nominal Young’s modulus (see Technical Details on page 9).

NOTE

From this point on, the strut works as a beam, so we will call it a beam, to make things clearer.

6. Put the beam onto the knife edge supports.

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7. Fit the digital indicator into its holder, so its display faces forward. The beam will bend downwards less than 10 mm in this experiment, so move the beam backwards temporarily. Adjust the deflection indicator in its holder to allow it to measure approximately 10 mm of downward beam deflection (see Figure 27).

10 mm of downward movement

Figure 27 Temporarily Push the Beam Behind the Tip and Adjust the Deflection Indicator for 10 mm Downward movement

8. Adjust the beam to be central across the Knife Edge Supports (so that an equal amount of beam ‘sticks out’ over the Knife Edge Supports) (see Figure 28). Or, use a pencil to mark the beam at its centre (375 mm) and adjust it so that the pencil mark is just under the tip of the Deflection Indic ator.

Equal ‘overhang’

888

Beam

Figure 28 Adjust the Beam So that an Equal Amount Sticks Out (Overhangs) Each End

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9. Hook the Knife Edge Weight Hanger onto the beam at the mid position, just under the tip of the Deflection Indicator (see Figure 29).

Knife Edge Weight Hanger

Figure 29 Fit the Knife Edge Weight Hanger to the Beam, Just Under the Tip of the Deflection Indicator

10. Zero the deflection indicator. The deflection reading from this point onwards ignores any initial bend in the beam and any bend caused by the small weight of the Knife Edge Weight Hanger. 11. If you have VDAS®, enter the distances between supports and the deflection indicator position. 12. Fit the Hooked Weight Hanger to the bottom of the Knife Edge Weight Hanger. Add 9 x 10 g weights to give a total of 100 g load*. Gently tap the frame to reduce the effect of friction. Record the reading of the deflection indicator. If you are using VDAS®, enter the load value and click on the ‘Record Data Values’ button.

NOTE

*The hooked Weight Hanger weighs 10 g, so you must allow for this.

13. Increase the load to 200 g, 300 g, 400 g and 500 g. At each increase, record the deflection.

R es u l t s A n a ly s i s Convert your load into force and if necessary, convert your deflection into metres (you can use mm or metres, but the correct SI unit is metres). Plot a curve of force (vertical axis) against deflection (horizontal axis) for the beam. What does your curve suggest about the behaviour of the beam? Use your measurements of the beam dimensions to calculate the second moment of area for the beam. Refer to the Technical Details on page 9 to find the Young’s Modulus for the material that makes your beam. Use the simple beam bending equation in the theo ry section of this guide to find the theoretical deflection of the beam for each load. Add your theoretical curve to the same chart as the actual curve and compare the two curves. Does the theory accurately predict the deflection of the beam? What are the possible sources of errorif any?

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Experiment 2 - Stiffness (Young's Modulus) of the Strut Materials Aims To test the struts and use your results to find their actual stiffness, and from this, find the Young’s Modulus for the material that makes the strut.

Notes In experiment 1 you use a nominal value of Young's Modulus for your calculations, which can cause errors, as the metal that makes your struts may not have a consistent Young’s Modulus. Accurate tests with this equipment will give you more useful and accurate values for the Young’s Modulus for your struts. You can then use these values for more accurate results in later experiments.

Procedure 1. Create a blank table of results, similar to Table 2.

Beam Material: Beam Width: Beam Thickness: Distance between supports ( L):

Load (g)

Force (N)

0

Second Moment of Area ( I ):

Measured Deflection (mm)

Measured Deflection (m)

48yI/L3

0

100 200 300 400 500 Calculated Young’s Modulus ( E ):


2. Repeat the procedure for experiment 1, using the 750 mm long steel, aluminium and brass struts.

R es u l t s A n a ly s i s Convert your load into force and if necessary, convert your deflection into metres (you can use mm or metres, but the correct SI unit is metres). For each strut, create a chart of force (vertical axis) against 48 yI/L3. Find the gradient of each curve to find the true Young’s Modulus for the metal used to make the strut. Compare your results to the nominal (textbook) values given in Young’s Modulus on page 10. Do they compare well?

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Further Experiments with Beams - Beam Deflection The Loading and Buckling and Struts apparatus works well to prove theory for simple beams on two supports. You can adjust the knife-edge supports to different positions and apply the load at different positions to create different beam experiments. You could also use the deflection indicator to measure deflection at different positions along the beam, to find the deflection profile of the beam.

Sugg ested Procedure to Measure Deflection Profile of a Beam 1. Choose your beam. 2. Set the knife edge supports to a suitable distance apart. 3. Check that your beam looks straight. Rest it on the supports. 4. Loosen the fixings for the deflection indicator and slide it to be as near as possible to one of the supports. Tighten its fixings and set it to zero. This gives you a zero datum. 5. Loosen, move and re-tighten the deflection indicator along the base in 25 mm steps. At each step, record the deflection and position. This gives you a datum profile for the unloaded beam. 6. Add the load to the beam. Do not reset the deflection indicator. Move it to the same 25 mm spaced positions you used in the last step and again measure the deflection. This gives you a profile for the loaded beam. Subtract the (datum) unloaded profile fro m the loaded profile to get a true deflection profile for the beam.

NOTE

User Guide

Take care how you adjust and slide the deflection indicator holder along to measure deflection. Always re-tighten the fixings before you take a reading. Do not leave the fixings loose and slide the holder along as you take readings, because you will get unreliable results.

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Experiment 3 - The Deflected Shape of a Strut A im To prove the theory about the sinusoidal shape of buckled struts, for each end condition (fixing).

Notes You will see immediately the sinusoidal shape of the strut as it buckles . Its amplitude will grow as the load increases towards the buckling load. So, in this experiment you add enough load to create a reasonable and measurable deflection. The actual load value is not important. You need two people to do this experiment correctly. One person to load the strut and record the positions for the deflection indicator, the other person to move the deflection indicator and record its deflection readings.

Procedure

NOTE

This procedure works for all struts and all ends conditions. TecQuipment recommend that you start with pinned - pinned ends condition and t he 600 mm steel strut.

1. Find the strut you need for your test and make a pencil mark at its mid-point (for example - make a pencil mark at 375 mm along a 750 mm strut). For reference, measure the thickness and width of the strut. 2. Connect and switch on the Load Display. Allow a few minutes for the display and the load cell of the measuring end to warm up. Tap the load measuring end to remove any effects of frictio n, then zero the display. 3. Turn the hand wheel of the loading end to give 5 to 10 mm gap behind its chuck (see Figure 30).

5 to 10 mm

Figure 30 Turn the Hand Wheel to Give 5 to 10 mm Gap Behind the Chuck

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4. Use the hexagon tool supplied, to loosen the four screws securing the loading end and slide it along the base until your strut fits into each chuck for the e nd condition you need, as shown in Figure 31. Re-tighten the four screws.

NOTE

* Note that for the fixed-pinned ends condition, you may need to use the 1 mm or 3 mm spacers in the chucks to keep the strut loading accurately aligned. This helps to prevent an eccentric fixing.

Pinned - Pinned Ends

Fixed - Fixed Ends

Fixed - Pinned Ends

*

Figure 31 Different End Fixings

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5. Fit the deflection indicator on its L-shaped holder, to the top of the base (see Figure 32).

NOTE

There are two holders that will hold the deflection indicator. You must use the Lshaped holder for the strut experiments.

6. Adjust it so its tip touches your pencil mark, half-way along the strut. 7. In this experiment, your strut will only bend by a maximum of 10 mm away from the Deflection Indicator, so adjust it in its holder, so that its tip will extend at least 10 mm when the strut bends.

NOTE

If the indicator has a +/- button, set it to show a positive number when it extends.

Adjust the Indicator to allow 10 mm of movement, this way (outwards).

Pencil mark at halfway along the strut.

Figure 32 Fit the Deflection Indicator to its L-shape Holder and Fit it to the Top of the Base

8. Create a table of results, similar to Table 3. If you are to use VDAS®, select the Strut Experiments. The software will create your results table for you automatically. 9. Use the large hand wheel to apply a small force to the strut. Check that its bends away from the deflection indicator. If not, reduce the force and turn the strut around. 10. Use the large hand wheel and carefully add a small load to the strut (less than 5 N). This helps to check the end fixings are holding the strut securely, especially if you are testing with the pinned ends condition. 11. Zero the deflection indicator reading.

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Beam Material: Beam Length: Beam Dimensions:

Deflection Position (25 mm steps)

End Fixing Conditions: Load:

Deflection Reading (Datum)

0 (Mid Point)

Deflection Reading (Loaded)

Actual Deflection (Loaded Datum)

0

+25 mm +50 mm +75 mm +100 mm +125 mm +150 mm Right (Positive)

+175 mm + 200 mm + 225 mm + 250 mm + 275 mm

-25 mm -50 mm -75 mm -100 mm -125 mm -150 mm Left (Negative)

-175 mm -200 mm -225 mm -250 mm -275 mm


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NOTE

Just as in the test with struts as beams, you must now find the unloaded (datum) shape of the strut and subtract this from the loaded shape to find the actual shape due to the load.

12. Move the indicator along the strut, from the halfway pencil mark towards the right in 25 mm steps. At each step, re-tighten the deflection indicator fixings before you take a deflection reading. If you are to use VDAS®, remember to record the deflection indic ator position and click the ‘Record Data Values’ button. 13. When you have reached as far right as you can go, set the indicator back to the halfway pencil marks and move in 25 mm steps to the left of the strut, recording the deflection results as negative values (as shown in the results table).

NOTE

Your last positions may not be an exact 25 mm, because of the tip of the deflection indicator, so just note the actual position.

14. Move the deflection indicator back to the halfway point. Use the hand wheel to load the strut until the central deflection reaches approximately 6 mm. Record the load for reference. 15. Repeat steps 12 and 13, recording deflected readings for the loaded strut. 16. Repeat the experiment with fixed - fixed end conditions and strut number 3. 17. Repeat the experiment with fixed - pinned end conditions and strut number 4. Use the loading end as the fixed end.

R es u l t s A n a ly s i s As shown in the results table for each test, subtract your unloaded (datum) results from the loaded results to get the actual deflection. Be careful with your signs when doing this. Plot a graph of deflection (vertical axis) against position along the str ut (horizontal axis). Make sure your horizontal axis has an equal negative and positive scale. For the pinned ends condition, over your results, draw a half sine wave of the same amplitude and cycle length. For the fixed ends condition, draw a full sine wave over you r results. For the fixed - pinned ends condition, draw a 3/4 sine wave over your results. Do your results match the theory for sine wave shapes?

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Experiment 4 - The Euler Buckling Load using Pinned-end Struts Aims To compare theoretical buckling load with actual buckling loads of pinned end str uts from experiments and prove the theory and show its limits.

Procedure 1. Create a blank results table, similar to Table 4. If you are to use VDAS®, select the Strut Experiments. The software will create a table of results for you automatically.

Strut Details

Length

Material

Second Moment of Area

Peak (Buckling) Load 1


Average Peak (Buckling) Load

Theoretical Buckling Load


2. Connect and switch on the Load Display. Allow a few minutes for the display and the load cell of the measuring end to warm up. Tap the load measuring end to remove any effects of frictio n, then zero the display. 3. Find the 750 mm steel strut. Use a micrometer or vernier and carefully measure its dimensions, and calculate its second moment of area. 4. Fit the strut in the pinned ends condition as described in Experiment 3 - The Deflected Shape of a Strut, but remove the deflection indicator.

NOTE

You must now buckle the strut, then buckle it again, in the opposite direction. This gives you two test results, and you find the average peak (buckling) load to get a good result. The first time the strut buckles, it buckles in its ‘natural’ direction.

5. Use the large hand wheel to load the strut slowly. As you turn the hand wheel, watch the load reading and the deflection of the strut. When you see that the load does not incre ase, but the strut is still deflecting, the strut has buckled. Record the ‘peak load’, shown in the Load Display. Release the load.

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6. Apply a light load, and gently push the strut to make it buckle the opposite way to your last test. Increase the load until the strut buckles, and record the peak load. 7. Repeat the test for other struts of the same cross-section and second moment of area , but different lengths.

R es u l t s A n a ly s i s For each strut, calculate the average peak (buckling) load. Plot a curve of the length (vertical axis) against the buckling load. Use the second moment of area to calculate the theoretical buckling load for each length and plot it on the same chart.

NOTE

As with all your results, use suitable units. SI units are best, but can give numbers with many decimal places.

Does the Euler theory predict the buckling load well? You will notice that when the strut buckles in it's natural direction that the load is lower. Why is that? (Hint - see to the assumptions made in the Euler theory). Your curve will be non-linear, so it is difficult to see errors. The Euler Bu ckling Formula shows that you can plot 1/ L2 against buckling load to give a linear plot that makes it easier to compare results. This also shows that bucking load is proportional to 1/ L2.

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Experiment 5 - Comparing Buckling loads with End Conditions Aims • To test a strut fixed with all three end conditions and prove the relationship between the buckling load and the end conditions. • To help show the ‘effective length’ principle.

Procedure 1. Create a blank results table, similar to Table 5. If you are to use VDAS®, select the Strut Experiments. The software will create a table of results for you automatically.

Second Moment of Area for the Strut:

Fixing Condition

Strut

Pinned - Pinned

5

Pinned - Fixed

4

Fixed - Fixed

3





2. Connect and switch on the Load Display. Allow a few minutes for the display and the load cell of the measuring end to warm up. Tap the load measuring end to remove any effects of frictio n, then zero the display. 3. Find the 600 mm steel strut (number 5). Measure its dimensions accurately and find its second moment of area. Fit it in the pinned ends condition as described in Experiment 3 - The Deflected Shape of a Strut, but remove the deflection indicator.

NOTE

You must now buckle the strut, then buckle it again, in the opposite direction. This gives you two test results, and you find the average peak (buckling) load to get a good result. The first time the strut buckles, it buckles in its ‘natural’ direction.

4. Use the large hand wheel to load the strut slowly. As you turn the hand wheel, watch the load reading and the deflection of the strut. When you see that the load does not incre ase, but the strut is still deflecting, the strut has buckled. Record the ‘peak load’, shown in the Load Display. Release the load. 5. Apply a light load, and gently push the strut to make it buckle the opposite way to your last test. Increase the load until the strut buckles, and record the peak load. 6. Release the load and remove the strut.

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7. Now find and fit strut number 4 in the fixed - pinned condition. Use the loading end as the fixed end. This strut is 25 mm longer than strut number 3, to allow for the length lost in one fixing. This gives a constant test length for correct comparisons. 8. Repeat the test and record the loads. 9. Repeat the test for strut number 3 in the fixed-fixed end condition. Again, the length of this strut allows for the length lost in the fixings, to give a fair comparison.

R es u l t s A n a ly s i s For each strut, calculate the average peak buckling load. Do the loads for each fixing condition follow the theory (fixed-fixed buckles at four times the load of pinned-pinned, and fixed-pinned buckles at twice the load of pinned-pinned). Your results for the fixed-fixed condition may be lower than you expect. Can you explain why? Think about the load on the fixings (chucks) as it buckles, and the effect it has on the end conditions. Add your results from this experiment to those of the last experiment (if you have done it). What do you think about the effective length idea?

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Experiment 6 - The Southwell Plot and the Buckling Load Aims To show how to use the Southwell plot to find the buckling load of a strut, and prove its usefulness.

Procedure 1. Create a blank table of results, similar to Table 6.

Load (N)

Deflection (mm)

0

0

Deflection /Load


2. Connect and switch on the Load Display. Allow a few minutes for the display and the load cell of the measuring end to warm up. Tap the load measuring end to remove any effects of frictio n, then zero the display. 3. Find the 600 mm steel strut (strut number 5). 4. Fit the strut in the pinned ends condition as described in Experiment 3 - The Deflected Shape of a Strut. 5. Use the large hand wheel to load the strut slowly to get a deflection of 0.5 mm. As you turn the hand wheel, gently tap the base to help remove any friction in the deflection indicator. Watch the load reading and the deflection of the strut. Record the deflection and load at approximately 0.5 mm intervals until you reach 4 mm deflection. Release the load.

R es u l t s A n a ly s i s Divide your central deflection ( yc) results by the load ( P ) at each deflection to complete your table. Create a Southwell Plot of central deflection ( yc) against ( yc /P ), in fundamental units. From this, note the gradient, to give you the buckling load of the strut. If you have done the earlier experiments, does the gradient agree with the theoretical and actual buckling loads for this strut?

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Experiment 7 - The Southwell Plot and Eccentricity of Loading Notes The last experiment showed that the gradient of the Southwell plot gave us the Euler buckling load and compared well with theory and actual experiment results. This plot can also show the effective eccentricity of loading, or out-of-straightness. As shown in the theory, imperfections in the strut cause the eccentricity of loading (also, the mechanical tolerances in the equipment may have a small effect). From this, if you know the eccentricity of each of a set of otherwise identical struts, the Southwell plots for each should have the same gradient, but the intercept should match the known eccentricity plus or minus the other imperfections.

Aims To show how to use the Southwell Plot to identify the eccentricity of loading.

Notes This test uses the longest strut (750 mm long) as the reference, with zero eccentricity. It then uses the next longest strut fitted with the special end fittings to mimic a strut with a known eccen tricity. The end fittings make this second strut the same length as the first, for a fair comparison. You test the second strut with the minimum and maximum eccentricity that the end fittings will allow. The deflection of each strut near to its ends will not be the same, due to the special end fittings. But because you use relatively long struts, the actual deflection at the ends is relatively small (compared to the deflection at the mid point of the strut), so the error is small.

Procedure 1. Create a blank table of results, similar to Table 7. 2. As shown in earlier experiments, set up the 750 mm long steel specimen strut (number 1) as a pinned - pinned strut. 3. As in Experiment 4 - The Euler Buckling Load using Pinned-end Struts , test this strut in both directions. Increase the load in steps of 0.5 mm deflection until it buckles (the load stops rising as fast, but deflection continues).

NOTE

For the longest strut, the maximum deflection will be less than 15 mm, so do not bend your struts more than this.

4. Fit the Eccentric End fittings to strut number 2, with both fittings set to give the smallest eccentricity (5 mm) (see Figure 6). 5. Fit strut number 2 and readjust the position of the deflection indicator to allow for the offset. 6. Repeat the test with this strut. 7. Reverse the end fittings at both ends to give the larger eccentricity, adjust the indicator again and repeat the tests.

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Eccentricity (0, 5 mm or 7.5 mm):

Deflection (mm)

Load 1(N)

Load 2 (N)

0 (0) 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0 16.5 17.0 17.5 18.0 18.5 19.0 19.5 20.0 20.5 21.0 21.5 22.0 22.5 23.0 23.5 24.0 24.5 25.0 25.5 26.0 26.5 27.0

0

0

Average Load

Deflection/ Average Load


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R es u l t s A n a ly s i s For each line of results, find the average load and deflection/load. Create one chart of load (vertical axis) against deflection. Add to this chart your results from each strut. Create one Southwell Plot chart. Add to this chart your results from each strut. Compare the results on the load against deflection chart. What effect does the offset have? Compare the Southwell plots, especially the gradients. Are the gradients similar? Has the eccentricity affected the buckling load? Compare the eccentricities, (the intercept on the y axis) to the actual known offset made by the s pecial end fitting (5 mm and 7.5 mm). Do the results confirm what we expect (plus or minus the other nonideal effects)? Other than the effects of the end fittings, examine the equipment and see if you can find an important part of its design that might affect eccentricity of loading on the struts.

NOTE

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The way you draw a line through your results on the Southwell Plot affects the intercept greatly. Take care to draw a straight line through only the most linear part of your results.

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Further Experiments The Loading and Buckling of Struts (SM1005) allows many more experiments than those suggested in this guide. For example - it has parts that allow more advanced study into the effects of additional lateral (side) loads. To apply a lateral (side) load, see Figure 33: 33: 1. Fit your your strut strut as descri described bed in in Experiment 3 - The Deflected Shape of a Strut. Strut . 2. Fit the pulley pulley into into the flat flat holder to the side side of the base. base. 3. Hook the knife edge weight weight hanger hanger with cord around the the strut. Lay the cord across across the pulley. pulley. 4. You You can now add a load to the cord cord to apply a side side load to the the strut.

Figure 33 How to Set Up a Lateral (side) Load.

Also, the extra specimens pack (SM1005A) (SM1 005A) will allow students to investigate: investigate : 1) The flexural rigidity and buckling loads for a further range of materials. 2) Tests on different engineering sections. 3) The effect of flexibility of end fittings. 4) The case of a compound (composite) Strut with imperfect shearing connections between the two components. Contact TecQuipment or your local agent for details.

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Results Note: These results are sample results only, actual results may be slightly different.

Experiment 1 - Deflection of a Simply Supported Beam

Steel Strut (Beam) Deflecti D eflection on 6 5

Actual Actual Defle ction Theory Deflection

) 4 N ( e 3 c r o F 2

1 0 0

0 .5

1

1 .5

2

2.5

Defle ction (mm) (mm)

Figure 34 Typical Typical Results For Experiment 1

Your results should show that the theory predicts the deflection accurately, while you deflect the beam within the elastic limits of its material. As shown in the theory, errors can be due to manufacturing tolerances in the material - its Young’s Young’s Modulus may not be accurate. accur ate. Also, the second moment of o f area may only be accurate for part of the beam, as its cross-sectional dimensions may change slightly along its length.

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Experiment 2 - Stiffness (Young's Modulus) of the Strut Materials

L Steel Strut (Beam) Force Against 48 yI / L

3

6 5 4 ) N ( e 3 c r o F

2

Slope = approx 200 GPa

1 0 0

5E- 12

1E- 11

1.5E-11

2E- 11

2.5E- 11

3E- 11

3 48 yI / L L

Figure 35 Typical Results For Experiment 2 - Steel Steel Strut

L Aluminium Strut (Beam) Force Against 48 yI / L

3

6 5 4 ) N ( e 3 c r o F 2

Slope = approx approx 68 GPa 1 0 0

1E-11

2E- 11

3E- 11

4E- 11

5E-11

6E- 11

7E- 11

8E- 11

3 48 yI / L L

Figure 36 Typical Results For Experiment 2 - Aluminium Strut Strut

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Brass Strut (Beam) Force Against 48 yI / L

3

6 5 4 ) N ( e 3 c r o F 2

Slope = approx 93 GPa 1 0 0

1E-11

2E-11

3E-11

4E-11

5E-11

6E-11

48 yI / L 3

Figure 37 Typical Results For Experiment 2 - Brass Strut

Your results should be similar to the nominal values, but will be more useful f or your later experiments. For reference, use a pencil to write your correct value for Young’s Modulus onto each of the beams you test.

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Experiment 3 - The Deflected Shape of a Strut

Deflection of Pinned - Pinned Strut against a Half Sine Wave 7 6 ) 5 m m ( 4 n o i t c 3 e l f e D

2 1 0 -350 -300 -250 -200 -150 -100

-50

0

50

100

150

200

250

300

350

300

350

Position Along Strut (mm)

Figure 38 Typical Results for a Pinned-Pinned End Strut

Deflection of Fixed - Fixed Strut against a Full Sine Wave 7 6 5

) m m4 ( n o i t c 3 e l f e D

2 1 0 -350 -300 -250

-200 -150 -100

-50

0

50

100

150

200

250

Position (mm)

Figure 39 Typical Results for a Fixed-Fixed End Strut

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Deflection of Fixed - Pinned S trut against 1/2 Sine Wave 7 6 5

) m m ( 4 n o i t c 3 e l f e D

2 1 0 -350

-250

-150

-50

50

150

250

350

Position Along Strut (mm)

Figure 40 Typical Results for a Fixed-Pinned End Strut

Your results should show that the theory accurately predicts the deflected shape of your struts.

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Experiment 4 - The Euler Buckling Load using Pinned-end Struts Strut Length against Buckling Load 800 750 ) m m700 ( t u r t s 650 f o h t g n 600 e L

Test Results Theoretical Results

550 500 150

200

250

300

350

Buckling Load (N)

Figure 41 Typical Strut Length against Load Results

1/ L Squared against Load 3.5 1/L Squared against actual results

3

1/L Squared against theory

) 2.5

2 -

m ( d 2 e r a u q 1.5 S L / 1

1 0.5 0 0

50

100

150

200

250

300

350

Load (N)

Figure 42 Typical Reciprocal Strut Length Squared against Load Results

Your results should be similar to the theory. The 1/ L2 chart should show the linearity of the length against load relationship.

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Experiment 5 - Comparing Buckling loads with End Conditions

Strut




Ratio

Pinned - Pinned

5

248

261

254.5

1

Pinned - Fixed

4

505

515

510

2

Fixed - Fixed

3

943

988

965.5

3.8

Fixing Condition

Table 8 Typical Results for Experiment 5

Your results should show the pinned - fixed buckling load to be twice t hat of the pinned - pinned buckling load. The fixed - fixed buckling load should be four times that of the pinned-pinned buckling load. However, the end fixings are not perfect, as they only use a simple mechanical clamp, so the ends of the strut might move slightly in the fixed-fixed condition (you would need to weld the fixings to the strut for a better fixing method). This will give a slightly lower than expected buckling load for the fixe d fixed ends condition.

Experiment 6 - The Southwell Plot and the Buckling Load Southwell Plot for 600 mm Steel Pinned - Pinned Strut 5 4 ) m m ( e r t n e c t a n o i t c e l f e D -0.005

Slope = 257

3 2 1 0 0

0.005

0.01

0.015

0.02

-1 -2 Deflection/Load (mm/N)

Figure 43 Typical Southwell Plot for the 600 mm Steel Strut with Pinned-Pinned Ends (I = 4.54 x 10 -11 )

This Southwell Plot gave a slope of 257 for a strut with a second moment area = 4.54 x 10 -11. This is very similar to the average buckling load of 254.5 N for this strut, found from earlier experiments, and also the theoretical buckling load of 261 N found in Experiment 4 - The Euler Buckling Load using Pinned-end Struts.

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Experiment 7 - The Southwell Plot and Eccentricity of Loading

Test on Struts with Three Different Eccentricities 180

Peak (buckling) load = approx 150 N

160 140


120 ) N ( 100 d a o 80 L


60

No fittings

40

Fittings set f or 5 mm eccentricity

20

Fittings set f or 7.5 mm eccentric ity

0 0

5

10

15

20

25

30

Deflection (mm)

Figure 44 Typical Load against Deflection for the Three Eccentricities

Southwell Plots for Three Different Eccentricities 30 No Fittings

25

Fittings set f or 5 mm eccentric ity Fittings set f or 7.5 mm eccentricity

20 ) m m ( n o i t c e l f e D

15

All slopes = approx 155 10 5 0

-0.05

0

0.05

0.1

0.15

0.2

0.25

-5 -10 Deflection/load (mm/N)

Figure 45 Typical Southwell Plots for the Three Eccentricities

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The load against deflection chart shows that the standard strut (no eccentricity fittings) is ‘stiffer’, because its initial rate of deflection is lower than the othe r two results. It also accepts more load than the other struts before it reaches its buckling load (when the curves become almost level). This shows that increasing eccentricity lowers the actual buckling load. The three gradients of the Southwell Plot show that the Euler buckling load is similar (the gradients are the similar) even though the eccentricity increases. The intercepts on the Southwell Plot are interesting. They show a ‘base line’ eccentricity of approximately 2 mm for the strut with no fittings. This baseline eccentricity is also present and in addition to the known eccentricities of the other two struts, to give intercepts of approximately 7 mm and 10.5 mm. This shows that the Southwell Plot is a reliable tool to help find the Euler buckling load of struts, even with large eccentricities. It also allows you to test a strut without danger of applying too much load and damaging the strut, as you do not need to test it to the full buckling load. Figure 46 shows the possible alignment errors (exaggerated for clarity) that might be present in the equipment, before or during loading, allowing for manufacturing tolerances.

Figure 46 Possible Alignment Errors (Exaggerated)

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Useful Textbooks Structural and Stress Analysis By T H G Megson Published by John Wiley & Sons ISBN 0 470 23563 2

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Maintenance, Spare Parts and Customer Care Maintenance General Regularly check all parts of the equipment for damage, renew if necessary. When not in use, store the equipment in a dry dust free area, preferably covered with a plastic sheet. If the equipment becomes dirty, wipe the surfaces with a damp, clean cloth. Do not use abrasive cleaners. Regularly check all fixings and fastenings for tightness; adjust where necessary.

NOTE

Renew or replace faulty or damaged parts or detachable cables with an equivalent item of the same type or rating.

Electrical You cannot repair the power supply for the Load Display. If it is faulty, replace it with a new and identical power supply.

WARNING

Do not open the power supply.

Fuse Location There are no user replaceable fuses on this equipment.

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SM1005 User Guide_0314.pdf

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